Adding Higher-Order Interactions to Network Ontology
Douglas R. White
Vladimir Batagelj
Andrej Mrvar
Balazs Vedres
Sept 1, 2004, SFI (1.00)
Network ontologies typically assume that network interaction is of the first-order,
at the level of interacting dyads. Here, we look at interactions where nodes interact
with dyadic edges, in various types of second-order interactions. We use the idea
of a tensor space of potential interaction among n-tuples but are careful to limit our
formalisms for such interactions to cases where such interactions are the most
plausible and the best documented. Our canonical examples are supervisory
relations (Nadel 1957) over co-workers or co-occupants of the same role, primate
interventions, catalysts that act on interaction rates between dyads, and industries
with horizontal or vertical integration of firms through common ownership
relations.
Second-order interactions may differ from ordinary triadic interactions in
various ways, as shown in Figure 1. Graph (a) shows an ordinary triad consisting
of all three possible pairwise relations among three nodes. Figure (b) is not
technically a graph but illustrates how a supervisor of two employees might
attempt to or succeed in exerting influence not just on the two subordinates
independently but on the dyadic edge or relationship between them. Figure (c) is
also not a graph but illustrates how a pair of individuals of opposite sex (e.g.,
married couple) might engender a child, and also have individual relationships
with the child. The basic idea here is that interaction with a pair of nodes is not
always reducible to interaction to each node in the pair individually.
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2
3
(a)
(b)
(c)
Figure 1. Ordinary Triadic versus Second-Order Node/Edge Interactions
It is the directed arrows between an individual and a pairwise relation that signal
the kinds of second-order interaction in which we are interested. We consider the
case where the receiving end of the arrow signals some sort of subordination, such
as child/parents or employees/supervisor, in a relationship structure in which a
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dyad may be superordinate (parents) or subordinate (employees) to one or more
other nodes.
Defining Second-Order Interaction as Join-Graphs
∩
To model second-order interactions as graphs we use Harary’s (1971) definitions
of digraphs, arcs and demiarcs. A digraph D=<V,A> is a set V of vertices or points
and a set of arcs or ordered pairs of distinct elements in V2. A demiarc is a point in
V together with a direction{>,<} for {out,in}. An out-demiarc u is denoted u> and
an in-demiarc v is denoted >v. A u,v arc may be composed of two demiarcs, u>
and >v, that join at u◊v. A demiarc thus has a stub or join at which it may join with
another demiarc to form an arc, and two demiarcs join to form an arc u,v when
they share a u◊v join.
Characterizing all arcs in a digraph as pairs of demiarcs with stubs is useful
for constructing a simulated digraph that has the same indegree and outdegree
distribution as one observed by disconnecting its demiarcs and rejoining them
according to some probability function.
A join-digraph D=<V,A,₪,J> is a digraph in which, in addition to nodes in
V and arcs in A
V2, there is a set ₪ of joins or stubs that join demiarcs in
(V,{>,<}).
We now generalize to the joining of demiarcs with a new graph structure
that allows joins to higher-order objects such as sets of interrelated elements. We
will have in mind the case where this kind of structure conveys interaction that
influences the edges or arcs among nodes rather than individually with the nodes
themselves. A good example of this process is catalytic action, in which one a
chemical substance X decays to Y at a rate that is governed not only by available
heat energy but the presence of a third, catalytic, substance Z.
A demitong is a set T of points in V together with a direction{>,<} for
{out,in}, and a k-demitong is one with cardinality k=|T|. An out-demitong t is
denoted t> and can join an in-demitong (or in-demiarc) denoted >u. A k-demitong
(1-demitong, 2-demitong, k-demitong), then, is a demitong directed towards k
nodes and their edges (arcs, loops). By this definition, while a 1-demitong is
directed towards a single node and thus has the structure of a demiarc, we might
consider it to be directed toward a loop of that node. The idea of a demitong is that
when it is part of a connection among nodes, it specifies how the nodes to which
the k contacts of a k-tong connect are simultaneously acted upon so as to affect the
relationships among the contact nodes. For this to occur, however, a demitong
must be joined with a matching demi-object to connect one or more nodes to the k
contacts of the k-demitong, which we call an activation.
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A tong-digraph is then a join-digraph D=<V,A,₪,J > in which in addition to
nodes in V and arcs in A, every arc has a corresponding pair of demiarcs, and there
are an additional set J of joined demiarc/demitong or demitong/demitong pairs
where the demitongs are ordered pairs in (Π(V),{>,<}) consisting of subsets in the
power-set of V ordered with respect to their joined demiarcs or demitongs by a
direction{>,<} for {out,in}. A tong-digraph is thus defined within a tensor
structure where Π(V) is the power-set of V, and it is possible to simulate a tongdigraph that has the same indegree and outdegree distribution as one observed by
disconnecting its demicomponents and rejoining them according to some
probability function.
A k-tong-digraph is one for which the largest cardinality of sets in its tensor
structure in Π(V) is k. A tong-digraph thus partitions the interactions in a network
into those of simple dyads conventionally represented by arcs (or, if symmetric,
edges), and potentially more complex interactions represented by joins with
demitongs. A 2-tong digraph is thus a tong digraph that contains at least one graph
structure composed of joined demiarc/demitong or demitong pairs (in addition to
demiarc pairs). In Figure 2, demitong joins are represented by the graphic symbol
. It shows the examples above diagrammed as 2-tong-digraphs as contrasted
with ordinary triads.
(a)
(b)
(c)
Figure 2. Second-Order Interactions as Graphs versus Ordinary Triads
Figure 2(c) shows an ordinary triad which, as in (a) and (b), may coexist with
higher order tong-structures. Figure 3 shows 2-demitongs and demiarcs as separate
or elementary structures, with the same graphic element for the joins of DTs and
Das. Figure 4 shows three ways that 2-demitongs may interlock with demiarcs or
with one another, with a doubling of the graphic element for the joins of two DTs.
(a) DT to pair
(b) DT from pair (c) DA to node
(d) DA from node
Figure 3. Elementary 2-Demitongs (DTs) and Demiarcs (DAs)
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(a) DT to a pair
(b) DT from a pair
(c) DTs to and from a pair
Figure 4. Interlocking DTs and DAs
A tong is thus an interlocked and ordered pair of demiarc/demitong relations
between a single node and one or more nodes and their arcs (edges, loops). Figures
4 (a) and (b) show tongs but 4(c) is more complex. It is important to distinguish
carefully the cases shown in Figure 5, where (a) shows two (out-) demiarcs joined
to a 2-tong, (b) shows a 2-tong joined to two (in-) demiarcs, and (c) is as before, a
2-in-tong jointed to a 2-out-tong. The difference is that in (a) the upper pair have
separate interests or influences on the relationship of the lower pair, while in (b)
the upper pair have common interests or influences on each of the lower pair
separately, while in (c) the upper pair have common interests or influences on the
relationship of the lower pair.
(a) DT to a pair
(b) DT from a pair
(c) DTs to and from a pair
Figure 5. Interlocking DTs
It is important to note that in the general case, a tong-digraph D=<V,A,₪,J > is not
a bipartite graph with two sets of nodes, V and J, because set J is one of
demiarc/demitong or demitong/demitong pairs that join to connect nodes in V but
according to different orders of interaction. The demiarc and demitong
construction of tong-digraphs, however, makes it possible to compare then to
simulated graph structures with the same indegree and outdegree distributions. In
some limited cases, however, it may be relevant and useful to define V and ₪ as
two distinct sets of nodes.
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Applying the Tong-Digraph Ontology
(1) Kinship
A bipartite p-graph is an example of a tong-digraph D=<V,A,₪,J > representing
kinship and genealogical connections in which V is the set of people, ₪ the set of
marriages and single (unmarried) individuals, J the mappings from V into ₪
according to who marries whom and from ₪ into V according to which couples are
the parents of individuals in V, and A are the derivative arcs from individual
parents to individual children. Figure 6 shows an example of such a bipartite
kinship graph for 4-marriages in which two brother-sister pairs intermarry. Not
shown are the derived parent-child arcs from the senior to the junior generation
within each nuclear family. A bipartite graph such as this has some affinity to a
Petri-net (refs) in which there are two types of nodes, one set (the square nodes for
example) aggregating resources from the other set (here, individual parents) to pass
down to other members of that set (individual chidren).
Figure 6. Tong-digraph for Kinship Networks
Bipartite p-graphs as diagrammed as tong-digraphs as in Figure 6 are highly
effective in representing large kinship networks and in finding cycles of
intermarriage within families and among sets of families.
(2) Role Interlock
Nadel (1957) argues that the coherence of interrelations among social roles in a
community or society is brought about by supervisory relationships in which some
individuals intervened to reshape the dyadic relationships of others. A doctor with
several nurses, for example, will intervene in a supervisory capacity to shape the
interactions among them, other supervisory interventions will occur between
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doctors and their patients, between patients and their insurance companies, and
between the latter and the set of doctors. In an organized community of interaction,
these interlocks will intermesh in ways that tend to create a coherent system of
norms, sanctions, and interpersonal expectations.
Figure 7 illustrates a situation of role interlock commonly observed in
societies in which marriage with a wife’s sister is allowed as a means of
reinforcing or continuing an alliance between families or descent groups. Here
there are now two sets of arcs, solid arrows representing actual kinship and
marriage links, and dotted arrows representing potential claims in secondary
marriages: a man with WiSi (sororal marriage), his brother with BrWiSi (extension
of the sororate), the wife with a departed HuBr (levirate), and so forth. As Nadel
noted in his discussion of kinship networks, it is in such cases that either (a) a man
avoids his wife’s parents or (b) jokes with his siblings-in-law. This situation of role
interlock is considered to be one of a potential conflict in supervisory authority: the
husband, for example, has a potential claim over an unmarried WiSi, while the
wife’s parents have a supervisory interest in the prospects for their unmarried
daughter’s marriage that includes the possibility of her not marrying the SiHi but
rather some other potential son-in-law from another family. Evidence that potential
role conflict is solved through coherent behavior interlock is the fact that either of
the possibilities (a) or (b) above almost invariably occur where sororatic or
leviratic claims are present between families, and that outcomes (a) and (b) occur
mutually exclusive of one another. The coherent logic here is that the potential
conflict is resolved if the parents-in-law are avoided, but if not, then it surfaces in
the form of ritualized joking between siblings-in-law, whose content is consistent
(in ways of joking about sex) with the unresolved ambiguity over the unmarried
WiSi/SiHu relationship.
Figure 7. Tong-digraph for Kinship Roles
(3) Alpha Primate Interventions in Juvenile Conflict Dyads
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In studies of primate behavior, recurrent conflicts in pairs or larger sets of juveniles
are easily and often observed and can be disruptive in a community if they escalate
into series or serious conflicts with disrupt activities of community members or do
them harm. Coding conflict dyads as demitongs and intervening adults as having
demiarc potential to couple with a demitong intervention that may dampen or halt
such conflicts has the potential, with appropriate use of simulating different
probabilistic models of demiarc/demitong couplings in the activation of such
interventions, to answer research questions such as:
What are the patterns of intervention as compared to different probability
models of demiarc joins with demitongs? What is the probability structure of
such joins for the adults, males or alphas? What is their probability structure
with respect to the juvenile conflict dyads?
What are the higher-order patterns, such as if one adult tends to intervene for
a given set of demitong pairs, will another adult’s probability of intervention
decrease for these pairs?
(4) Catalytic Action
Catalytic action occurs not only among threesomes of chemical interactions but in
actions that affect interaction rates between dyads in a variety of different network
contexts.
(5) Effects of Co-Ownership
industries with horizontal or vertical integration of firms through common
ownership relations.
The shift is from mundane relations among individuals to the higher order
(6) Three-Way Interaction Statistics
White, Pesner (1983)
Conclusions… (to be continued)
The substantive instantiation of a tong is the class of (often observable events of)
supervisory (interventionist) relations. By itself, a tong, simply by its existence has
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no effect unless it is at some point activated by one (or more) supervisors. After a
supervisory event operating through a tong has been activated one or more times,
and perhaps habituated, it may become internalized so that the (supervised) parties
to the tong may modify their dyadic behaviors in anticipation of the potential
activation of supervision.
A network with tongs is one with ordinary nodes, actors with dyadic interactions,
and k-tongs that are activators of events that alter relations among other sets of
nodes.
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White, D. R., R. Pesner, and K. Reitz. 1983. An Exact Significance Test for ThreeWay Interaction. Behavior Science Research 18:103-122.
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